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Electro-Magnetic
Guitar Pickups

Some Electronic Characteristics

(Jan 2003)

The basics of an electro-magnetic guitar
pickup is that it is a highly inductive coil placed in the near vicinity of a
constant magnetic field, that incidentally also has magnetic guitar strings in
that field.Any relative movement
of the magnet, and strings that causes a change in the reluctance of the
magnetic circuit will induce a voltage into the coil.This part of the study is focussed on the physical characteristics of the
coil (or coils) in a few sample pickups.

Because of the inter-winding
capacitance, and the high self-inductance due to the huge number of turns on
these pickups, the coils have a relatively low self-resonant frequency. Also the
resistance is relatively high in comparison to the inductance.To get a better understanding of these pickups, there are
some basic applied mathematical formulas that can characterise the pickups.

As these pickups are basically highly
inductive coils, there are two simple electrical measurements that can be
performed to measure the resistance and the inductance and these both give a
clear picture of the pickup coil and its electrical properties.

Resistance Factor

The resistance factor (Ar) is entirely
dependent on the geometry of the bobbin and the filling of the bobbin with
copper wire by a specified number of turns.The wire diameter (and insulation) limits the number of turns that can be
fit onto the bobbin, and that in turn directly relates to the resistance.The resistance is related to the resistivity of the wire, which relates
to the square of the cross sectional area of the wire strand.

Hence the very simple formula

R (micro ohms) = n^2 * Ar (micro-ohms)

This formula is fairly useful if you are
engineering the design of a coil, as you will have previously
determined the number the number of turns, and this then gives an expected
terminal resistance value – as you should already know the Ar value for that
bobbin.

Most coil bobbins are constructed in a
regular cylinder or square rectangular cylinder, but in the case of guitar
pickups, the shape is usually very flat, and can usually be seen as two
semi-circles joined by an extended straight section.It is important to understand this structure, as it is definitive in
formulating the resistance factor.

With a circular cylindrical bobbin
structure, the wire winds on over itself, and as it does, the radius
incrementally increases.This
incremental increase in the radius causes the length per turn to also
incrementally increase in a compound fashion and this is compound increase in
length is directly proportional to the resistance per turn.The formula for compound interest (Amount = Principal * Interest ^ Term
in years) follows this same structure.

In reverse, as this is a power
(geometrical) relationship, the mean radius is the square root of the inner
radius multiplied by the outer radius of the windings (excluding insulation
tapes).

Having now realised that the physical
structure of the pickup bobbin is not a round cylinder, it can be better
understood as two parts brought together:

Resistance = Ar.n^2 + Ax.n

In this case the Ar part related to the
circular ends and the Ax part related to the straight part of the winding
(bobbin).The number of turns
associated with the Ax factor is linear (order 1) because the length is
consistent irrespective of the number of turns on the bobbin.

So with two bobbins of different lengths
(x), between end centres then with the same number of turns with the same type
wire the Ar part will be constant – leaving the Ax part to be simply
calculated with simultaneous equations!This
leaves the Ar part to then be calculated for that same bobbin type – and this
then gives a predictive equation for coil resistance before the coil is
manufactured!

We don’t need to get into the depths
of this to better understand pickups, simply measuring the resistance is a good
starting point. What we do now understand is that the internal resistance
is very closely related to the square of the turns the bobbin, and that the
bobbin by its physical shape (specifying the cross sectional winding area, and
the mean length of the winding) totally defines the resistance (Ar)
factor.

This branch of applied mathematics
involves considerable approximation because there are so many variables that
surface - like bobbin filling factor (the proportion of the bobbin that is
filled with windings), wire insulation factor (the insulation thickness compared
to the wire diameter), winding factor (the total winding cross sectional area
compared to the total conductor cross sectional area), etc. All these and
other considerations can be included to obtain a far more accurate approximation
of the total resistance, but the range of available wire diameters is limited,
and this diameter varies - simply because the diamond drawing die wears with use
- so it is reasonable to expect considerable variation in resistance in
production runs, and my guess is that about +/- 5% variation would be reasonable
for production runs, and +/- 10% between production runs.

While these variations might seem quite
wide, it has to be understood that the bobbin is not round, and the winding
tension is really like a series of pulses as the ends of the bobbin come around,
so the wire gets stretched as it is wound on and this too will vary the
resistance, and the thickness of the total winding - as these impact on the
winding factor too! With different production runs the winding speed may be
different, tension different, wire source from another batch, enamel insulation
thickness different - so there is lots of room for production variation -
especially as this wire is so thin and there are so many turns, and the bobbin
is not a regular (round) shape for winding!

Inductance Factor

The Inductance of a coil (L) is
entirely dependent on the geometry of the magnetic circuit and the magnetic
components together with the number of turns in the coil.
In the same fashion as the Resistance Factor (Ar) is related to the physical
structure of the bobbin, the Inductance Factor (Al) is totally related to the
magnetic resistance (Reluctance) of the magnetic circuit, which is totally
dependent on the physical shape of the magnetic circuit.

These two entities Ar and Al fit like two pieces of a metal
chain, where each is an entity all by themselves, but together they form an
electro-magnetically coupled circuit!

Reluctance
is far more difficult to calculate than resistance, as the magnetic field is
very leaky, while with resistance the current that flows through a wire has very
clear boundaries making resistance relatively easy to approximate, the magnetic
flux that flows in a magnetic circuit passes through air, wood, plastic, brass,
iron, nickel etc. and all of these materials have differing permeabilities and
solving these problems usually reverts to rather complex mathematics called
Field Theory, and another associated branch of mathematics called Finite Element
Analysis.

For a known magnetic circuit the formula follows a similar shape as the
resistance factor.

Inductance (nH) = n^2 * Al (nH)

Again this formula is fairly useful if
you are engineering the design of a coil, as it will give you the expected
inductance value from the number of turns – as you would already know the Al
value from that magnetic construction (coil and magnet assembly) from earlier
measurements of a similarly constructed coil and magnetic components. So
in practice it would be the case of first wind a reference coil - with known
number of turns, assemble the magnetic structure and measure the inductance,
then deduce the inductance (Al) factor from
the above formula.

As with the resistance factor and its
association in a circular cylinder, the inductance factor follows the same
geometrical (power) relationship – but inductance is also proportional to the
square of the mean area defined by the straight length (x) and the mean radius
(y), so the more complete formula for inductance is:

Inductance = Al * n^2 + Am(x) * n * Am(y)
* n

Inductance = Al * n^2 + Am(xy)* n^2

Inductance= Al * n^2by including Am(xy) into Al

The Al factor is not that hard to
deduce, once a bobbin is filled and the inductance measured and the maths done
against the number of turns.We now
have a direct way to predict the inductance with reasonable accuracy before we even start
winding any pick up coils. Just as before with the resistance factor being determined by many
approximated sub-factors, so too some of these similar sub-factors play a part
in determining the inductance (Al) factor.

Wheeler's
Formula (and its extensions) go a long way to relate the physical shape of a
coil to the expected inductance and all of this is approximation theory at its
best! Wire diameters and bobbin filling factors are other issues
that will play big parts here as the (enamel) wire insulation is comparatively
thick but that is not in this scope.

What has to be realised here is that the
bobbin cross sectional magnetic path area is a vital component of the
Reluctance of the magnetic circuit and if this area is changed then it will have
a major effect on the overall reluctance, and this will have a direct effect on
the inductance. As the sides of the magnetic pickup are usually flat, then
they can be squeezed, and this could change the inductance - but we are not
going there!

The magnetic reluctance path is through
the magnet(s) then through the air and partially through the strings, then
through the air again all the way round to the distant end of the magnet.
Now a few home truths!

If the magnets were made a little bit
bigger/longer (like in the Telecaster) then there is more surface area for the
magnet field to launch itself compared to the air, and consequently the
reluctance is less, and the comparative inductance is greater for a similar
number of turns. For Hum Buckers, there are two coils in series, and these
have their magnets, and magnetic joining plate, and pole pieces to draw the
magnetic field around towards the strings. You would think that with all
this, that the inductance would be far greater and that the output level would
be far louder - or "hotter" in guitar speak!

Initial Measurements

For consistency, all initial inductance
tests were carried out at 1 kHz and the resistance was initially measured at 100
Hz with a HP 4262A LCR bridge.

The physical structure of these pickup
coils are reasonably similar, with a series of (pseudo) bar magnets through the
centre of the pickup coil, and the bobbins have a fairly consistent shape
between each other, and the resistances are fairly close at about 6.0 k ohms.It would be reasonable to expect that the number of turns
would be fairly consistent between these four pickups, so the inductance is the
variable!

It became obvious from the results below
that measuring the resistance must be done with a direct current, as even at 100
Hz a considerable error can exist.A
100 uA DC current source with a DVM gave reliable results. This is a low
current, and the resultant field (about 1 AT) is barely enough to affect the
permanent magnets.

The inductance was measured at 1 kHz
with the same HP 4262A LCR measuring bridge, and the results were tabulated
below:

Bobbin
Description

Resistance
(k)
HP100 Hz

Resistance
(k)
DC

Inductance
(H)

Strat 01

6.09 k

5.87 k

2.76 H

Strat 02

6.67 k

6.48 k

2.93 H

Kinman Strat

6.08 k

2.96 H

Kinman Tele

6.89 k

3.45 H

White Single

7.49 k

6.73 k

4.44 H

Black Single

5.44 k

5.12 k

3.05 H

HMB-01

Additive

8.72 k

8.45 k

4.59 H

Subtractive

8.35 k

8.45 k

3.72 H

Gr-Wh

4.30 k

4.33 k

2.07 H

Cm-Bl

4.13 k

4.12 k

2.08 H

HMB-02

R-Shield

15.48 k

14.3 k

9.05 H

R-W

7.42 k

7.18 k

3.92 H

W-Shield

7.31 k

7.16 k

3.84 H

These results showed that even as low as
100 Hz the reactive inductance had an appreciable effect on the impedance of the
pickups, but the resistance figures were consistent, indicating that these
resistance values were reliable and repeatable.

Now that the resistances and inductances
have been measured for these coils, and we have relationships that tie
resistance to the bobbin construction, and inductance that ties to the magnetic
construction, we are in a position to characterise various pickups in relation
to these measurements.

Quality Factor

For a coil, if an exciting current is
introduced, then current will flow through the coil, and the current will lag
the impressed voltage by an angle determined by the resistance and the reactance
of the coil.As the inductance is
measured at 1000 Hz, it makes sense to use this frequency as the measurement
base.

The “Quality Factor” or
"Q Factor" is an easy
method to determine the magnetic efficiency of a coil (at a specified
frequency), and it is simply the ratio of the inductive reactance (at 1000 Hz in
this case), divided by the resistance.Coils
with tight magnetic circuits can have the Quality Factor well exceeding 100 in
the audio range.The formula for
Quality Factor is:

In the table below the Q Factor was
derived by putting the measured inductance and resistance figures into the
equation and this gave the nominal Q Factor value for each pickup coil.

Bobbin
Description

Resistance
(k) DC

Inductance
(H)

Q
Factor
at 1 kHz

Cut
in Freq
(Hz)

Strat 01

5.87 k

2.76 H

2.99

338 Hz

Strat 02

6.48 k

2.93 H

2.92

352 Hz

Kinman Strat

6.08 k

2.96 H

3.05

327 Hz

Kinman Tele

6.89 k

3.45 H

3.15

318 Hz

White Single

6.73 k

4.44 H

4.14

241 Hz

Black Single

5.12 k

3.05 H

3.74

267 Hz

HMB-01

Additive

8.45 k

4.59 H

3.41

293 Hz

Subtractive

8.45 k

3.72 H

2.77

361 Hz

Gr-Wh

4.33 k

2.07 H

3.00

333 Hz

Cm-Bl

4.12 k

2.08 H

3.17

315 Hz

HMB-02

R-Shield

14.3 k

9.05 H

3.98

251 Hz

R-W

7.18 k

3.92 H

3.43

292 Hz

W-Shield

7.16 k

3.84 H

3.37

297 Hz

These figures showed up a few
interesting points.In general the
Q factor was in the range 3 to 4, with the White Strat having the highest Q
factor – it also has the longest pole pieces, a fuller wound bobbin than the
Black Strat and biggest pole spans.Strat
02 had the lowest Q factor, and it also had a two-third full bobbin compared to
the Strat 01 bobbin.

Individual measurements of coils for the
lateral Hum Buckers proved very interesting as the coils as entities all showed
similar figures to that of single coil strat coils - as expected.When the bucker coil was added in series the resistance added as
expected, but the inductance did not dramatically change when one of the coils
was reverse connected and this indicated that the mutual coupling (of the
magnetic fields) between these two coils was rather low.
This needs a little further investigation.

Mutual Inductance

When there is more than one coil, there
is coupling between the two coils and this is mutual inductance.In the case of the side by side (or laterally constructed) HMB-02 there are 3 inductance
measurements, one for each coil separately L1 and L2 and the combined one in
series L(tot). These three figures align with the three inductance values
in the following equation and from that, it is very easy to deduce the mutual
inductance.

L(tot) = L1 +
L2 +/- 2*M Henries

Or M = 0.5*(L(tot) +/– (L1 + L2))Henries

Substituting for the first one

M = 0.5*
(9.05 – (3.92 + 3.84)) Henries

So M = 0.645 Henries

Now, the coupling factor (k) is the
ratio of the mutual inductance divided by the mean inductance, and in a coil
that has a high coupling factor, then k approaches 1 (unity).

In this case, the coupling factor is
quite low. In other words, the two coils are rather poorly coupled, and
this is not surprising, as there is a big gap between the adjacent pole pieces,
and the pole pieces are spread (isolated) from each other - so that the
vibrating strings can be a prominent part of the magnetic circuit.

For the second coil set HMB -01

M = 0.5*(4.59 – (2.06 + 2.08))Henries

M = 0.225 Henries

And k =|M|/sqrt(L1*L2) = 0.109
again very low coupling.

With one of the coils reversed we get:

M = 0.5 * (3.72 + (2.07 + 2.08))Henries

M = 0.215Henries

So k = |M|/sqrt(L1*L2) = 0.104 which is consistent with above.

Again, ideally perfect coupling is where k = 1,
but this arrangement has to be loose as the strings need to be in the magnetic
circuit, and it is the string’s position that sets the instantaneous
reluctance and that in turn sets the flux density and it is the change in flux
density that causes a voltage to be induced into the windings.So mutual coupling is important if the lateral hum bucker is to have a
relatively closed magnetic field.

Cut-In Frequency

Another way to look at the ratio of
resistance to inductive reactance, is to locate the frequency where the
inductive reactance equals the winding resistance. In the low end of the audio spectrum
there will be a frequency where the pickup’s inductive reactance will equal
the pickup’s resistance and this can be directly related from Xl =
2 Pi f L, where the value of Xl is taken as the dc resistance R.

So f = R / (2 pi L)kHz(R
is in k ohms)

Actually the Cut-in Frequency is really
a one figure physical description of the structure of the pickup assembly, given
as a ‘Frequency Value’ which really has very little bearing on the actual
sound or timbre qualities that the pickup will give – that is much later!

From earlier measurements we knew that:

R=Ar*n2ohms

L=Al*n2nano henries

Substituting these physical Inductive
and Resistive Factors in place of R and L without the common turns value (n)
into this equation for R (Ar) and L (Al), the formula takes on a new light as:

Ar = 2 Pi f Al

or f = Ar/(2 Pi Al)kHz

This means that virtually any pickup can
be quantified by a couple of simple measurements, a little bit of maths and a
look up table, and we don’t need the number of turns to work things out!This does not lead anywhere particularly other than it shows
that there are general physical relationships that associate with the pickup
structures – and the structures are associated by their nature as a transducer
for moving magnetic strings.

Resonance Tester

By using a pair of Operational
Amplifiers in a positive feedback arrangement, stabilised by a thermistor, it is
quite easy to place a pickup coil in as a feedback component, and the circuit
will resonate at very near the resonant frequency of the pickup coil. The
22 pF is a phase corrector that brings the error into less than 3deg at 20
kHz. This little oscillator circuit puts out about 4 volts peak to peak,
and is a nice clean sine wave! The RA53 is a thermistor that lowers
its resistance as it warms up, and because the time constant is several seconds,
this keeps the output voltage quite constant over a range of pickups.

Here are some figures that describe some
pickups in terms of Resistance and Inductance and their Resonant frequencies:

Bobbin
Description

Resistance
(k)

Inductance
(H)

Frequency
(kHz)

Cap
(pF)

Strat 01

5.87 k

2.76 H

11.10 kHz

74.5 pF

Strat 02

6.48 k

2.93 H

9.90 kHz

88.2 pF

White Single

6.15 k

4.44 H

9.32 kHz

65.7 pF

Black Single

4.63 k

3.05 H

10.67 kHz

72.9 pF

Kinman Strat

6.08 k

2.96 H

9.91 kHz

87.3 pF

Kinman Tele

6.93 k

3.45 H

6.71 kHz

163.0 pF

HMB-01

Aiding

7.69 k

4.59 H

11.63 kHz

40.7 pF

Opposing

7.69 k

3.72 H

13.80 kHz

35.7 pF

Coil 1

3.99 k

2.07 H

14.93 kHz

54.9 pF

Coil 2

3.78 k

2.08 H

14.97 kHz

54.4 pF

HMB-02

R-Shield

12.72 k

9.05 H

6.184 kHz

73.2 pF

R-W

6.6 k

3.92 H

12.74 kHz

39.8 pF

W-Shield

6.55 k

3.84 H

6.383 kHz

162. pF

The Capacitance (C) is calculated from the resonant
frequency and the self inductance from the formula.

C = 1 /((2*PI*f)^2 *L)

C = 1000000 / (39.478 * F * F *
L) C in pF L in
Henries F in kHz

Do not be mislead by these results!
In theory because of the extra resistance in the coils, the resonant frequency
will be down by about 5% but in practice this does not matter, as there are
several other issues.

What has to be realised is that the resonant frequency is the point where
the distributed internal capacitance in the winding forms a parallel resonance
with the distributed internal self-inductance in the winding and that from
slightly below this frequency and forever above this frequency the output of the
pickup is severely limited. A parallel
resonance is very high impedance and therefore the output will be weak at and
above this resonant frequency. This is
a very powerful concept to comprehend and it is a prime factor in determining
the response of a guitar pickup.

A cursory glance over these figures
shows that main coil for the HMB-01 (W-Shield) has a self-resonance of about 6
kHz while its associate coil has a resonance of about 13 kHz, which is a bit
more than an octave above.This
seems wrong and the calculated self-capacitance is about 162 pF, and not in the
range of about 40 pF.All the other
self-capacitances are about 40 pF to 70 pF.It may have an electrostatic shield over it to shield it from
electrostatic noise and that could significantly increase the apparent
self-capacitance.

In and case, the load placed by a cable
is highly capacitive (about 47 pF per metre) so a 10 m cable can add about 470
pF to the apparent self-capacitance and really limit the frequency response!To further compound problems the volume control and tone controls all
interact to compromise the spectral output of a pickup and this needs to be
better understood.

Conclusions

All these pickup coils are of a rather
similar construction and all consist of an inductive winding that has
appreciable resistance – because of the thinness of the wire.Most of the coils have a resistance of about 4000 to 7000 ohms per coil
and the inductance is about 2.7 to about 4.4 Henries per coil.

With the Humbuck designed coils, these
were added in series to the initial coil and although the resistance doubled –
as expected, the inductance did not increase by the square of the total turns
– even thought the magnetic circuits were coupled.Humbuck coils are more loosely coupled than the Single Coil designs (which
are also loosely coupled). This brought in the thought of loose coupling and with that a poor Quality
factor (inductive reactance compared to resistance), and in most cases the Q
Factor was in the order of 3 to 4, which was a far cry from coils with a much
smaller and less lossy magnetic circuit having Q Factors well beyond 100 at 1
kHz.

Looking at inductive reactance another
way, there will be a frequency where the inductive reactance (at 1 kHz) will
equal the resistance, and this in general was about 250 to 350 Hz.In other words, these pickup coils can really be seen as inductive energy
sources and not resistive energy sources over much of the audio spectrum.This is a very significant point as it means that the load (volume
control, tone control amplifier, cable etc) will have a spectral effect on the
response – even if they are resistive!

As the mutual inductance was calculated
to be quite low, it then followed on that the coupling factor between the two
hum bucking coils was also rather low (somewhere between 0.16 and 0.20), and
this may seriously impinge on the ability of this physical arrangement to be effective!

In testing the self-resonance it was
then possible to calculate the effective self capacitance of the coils, and this
was in general about 40 to 70 pF, with self resonances about 9 kHz to15 kHz.Although this would appear to put the problem of self-resonance outside
the audio band, it has to be realised that the cable is highly capacitive, as is
the tone control, and these may have serious spectral limiting effects on the
overall output.The spectral response
from these types of pickups is very highly controlled by the electronic load
placed on the terminals of the pickup.

Now that we have a basic understanding
of the external electrical characteristics, these can be used to develop a model to
approximate the spectral response of the pickups in a unified manner when being
used as a signal generator!